R E S E A R C H
Open Access
Superconvergence of semidiscrete finite
element methods for bilinear parabolic
optimal control problems
Yuelong Tang
1,2*and Yuchun Hua
1*Correspondence:
[email protected] 1Institute of Computational
Mathematics, College of Science, Hunan University of Science and Engineering, Yongzhou, Hunan 425100, China
2Office of Continuing Education,
Peking University, Beijing, 100871, China
Abstract
In this paper, a semidiscrete finite element method for solving bilinear parabolic optimal control problems is considered. Firstly, we present a finite element approximation of the model problem. Secondly, we bring in some important intermediate variables and their error estimates. Thirdly, we derive a priori error estimates of the approximation scheme. Finally, we obtain the superconvergence between the semidiscrete finite element solutions and projections of the exact solutions. A numerical example is presented to verify our theoretical results.
MSC: 49J20; 65M60
Keywords: superconvergence; finite element method; bilinear parabolic optimal control problems
1 Introduction
We consider the following bilinear parabolic optimal control problem:
min
u∈K
T
y(t,x) –yd(t,x)L()+u(t,x)
L()
dt, (.)
∂ty(t,x) –div
A(x)∇y(t,x)+u(t,x)y(t,x) =f(t,x), t∈J,x∈, (.)
y(t,x) = , t∈J,x∈∂, (.)
y(,x) =y(x), x∈, (.)
where∈Ris a convex polygon with the boundary∂, andJ= [,T] ( <T< +∞). The
coefficient matrixA(x) = (aij(x))×∈[W,∞(¯)]×is a symmetric and positive definite.
Moreover, we assume thatf(t,x)∈C(J;L()),y(x)∈H(), and the admissible control
setKis defined by
K=v(t,x)∈LJ;L():a≤v(t,x)≤b, a.e. inJ×,
where ≤a<bare real numbers.
There has been a wide range of research on finite element approximation of elliptic op-timal control problems. For finite element solving linear and semilinear elliptic control problems, a priori error estimates were investigated in [] and [], and superconvergence
were established in [] and [], respectively. Yang et al. [] obtained the superconvergence of finite element approximation of bilinear elliptic control problems. In addition, some similar results of mixed finite element approximation for linear elliptic control problems can be found in [, ].
In recent years, there are a lot of related works on finite element approximation of parabolic optimal control problems, mostly focused on linear or semilinear cases. A pri-ori error estimates of space-time finite element and standard finite element approxima-tion for linear parabolic control problem were derived in [] and []. The superconver-gence of variational discretization and standard finite element approximation for semilin-ear parabolic control problem can be found in [] and [], respectively.
As far as we know, there has been little work done on bilinear parabolic control prob-lems. In this paper, we purpose to obtain the superconvergence properties of semidiscrete finite element method for bilinear parabolic optimal control problems.
We adopt the notation Wm,q() for Sobolev spaces on with norm ·
Wm,q() and
seminorm | · |Wm,q(). We set H
()≡ {v∈H() :v|∂ = } and denote Wm,() by
Hm(). We denote byLs(J;Wm,q()) the Banach space of allLsintegrable functions from
J into Wm,q() with norm v
Ls(J;Wm,q())= (T
vsWm,q()dt)
s for s∈[,∞) and the
standard modification for s=∞. Similarly, we can define the spaceHl(J;Wm,q()) and
Ck(J;Wm,q()) (see e.g. []). In addition, letcorCbe generic positive constants.
The rest of this paper is organized as follows. A semidiscrete finite element approxima-tion of (.)-(.) is presented in Secapproxima-tion . Some important intermediate variables and their error estimates are introduced in Section . In Section , a priori error estimates of the approximation scheme are derived. In Section , the superconvergence between pro-jections of the exact solutions and the finite element solutions is obtained. A numerical example is presented to illustrate our theoretical results in the last section.
2 A semidiscrete finite element approximation
We now consider a standard semidiscrete finite element approximation of (.)-(.). To ease the exposition, we denoteLp(J;Wm,q()) and · Lp(J;Wm,q())by Lp(Wm,q) and
· Lp(Wm,q)respectively. LetW =H() andU=L(). Moreover, we denote · Hm()
and · L()by · mand · , respectively. Let
a(v,w) =
(A∇v)· ∇w, ∀v,w∈W,
(f,f) =
f·f, ∀f,f∈U.
From the assumptions onAwe have
a(v,v)≥cv
, a(v,w) ≤Cvw, ∀v,w∈W.
The weak formulation of (.)-(.) can be read as follows:
min
u∈K
T
y–yd+u
dt, (.)
(∂ty,w) +a(y,w) + (uy,w) = (f,w), ∀w∈W,t∈J, (.)
It follows from (see e.g. []) that problem (.)-(.) has at least one solution (y,u) and that if the pair (y,u)∈(H(L)∩L(H
))×K is a solution of (.)-(.), then there is a
costatep∈(H(L)∩L(H
)) such that the triplet (y,p,u) meets the following optimality
conditions:
(∂ty,w) +a(y,w) + (uy,w) = (f,w), ∀w∈W,t∈J, (.)
y(,x) =y(x), ∀x∈, (.)
–(∂tp,q) +a(q,p) + (up,q) = (y–yd,q), ∀q∈W,t∈J, (.)
p(T,x) = , ∀x∈, (.)
T
(u–yp,v–u)dt≥, ∀v∈K. (.)
As in [], it is easy to get the following lemma.
Lemma . Let(y,p,u)be the solution of(.)-(.).Then
u=minmax(a,yp),b. (.)
LetP be the space of polynomials not exceeding , andTh be regular triangulations
ofsuch that¯ =τ∈Thτ¯andh=maxτ∈Th{hτ}, wherehτ denotes the diameter of the
elementτ. Furthermore, we set
Uh=
vh∈L() :vh|τ= constant,∀τ∈Th
,
Wh=
vh∈C(¯) :vh|τ∈P,∀τ∈Th,vh|∂=
.
As in [], we assume that
Kh=
vh∈Uh:a≤vh|τ ≤b,∀τ∈Th
is a closed convex set inUh. We recast a semidiscrete finite element approximation of
(.)-(.) as
min
uh∈L(Kh)
T
yh–yd+uh
dt, (.)
(∂tyh,wh) +a(yh,wh) + (uhyh,wh) = (f,wh), ∀wh∈Wh,t∈J, (.)
yh(,x) =yh(x), ∀x∈, (.)
whereyh(x) =Rh(y(x)), andRhis an elliptic projection operator, which will be specified
later.
It is well known that (.)-(.) again has a solution (yh,uh) and that if the pair (yh,uh)∈
H(W
h)×L(Kh) is a solution of (.)-(.), then there is a costateph∈H(Wh) such that
the triplet (yh,ph,uh) meets the following conditions:
(∂tyh,wh) +a(yh,wh) + (uhyh,wh) = (f,wh), ∀wh∈Wh,t∈J, (.)
–(∂tph,qh) +a(qh,ph) + (uhph,qh) = (yh–yd,qh), ∀qh∈Wh,t∈J, (.)
ph(T,x) = , ∀x∈, (.)
T
(uh–yhph,vh–uh)dt≥, ∀vh∈Kh. (.)
We introduce the averaging operatorπc
hfromUontoUhas
πhcv|τ=
|τ|
τ
v dx, ∀τ∈Th, (.)
where|τ|is the measure ofτ. Then we can similarly derive the following lemma.
Lemma . Let(yh,ph,uh)be the solution of(.)-(.).Then we have
uh=min
maxa,πhc(yhph)
,b. (.)
3 Error estimates of intermediate variables
In this section, we introduce some important intermediate variables and derive some re-lated error estimates. For allv∈K, lety(v),p(v)∈H(L)∩L(H) satisfy the following equations:
∂ty(v),w
+ay(v),w+vy(v),w= (f,w), ∀w∈W,t∈J, (.)
y(v)(,x) =y(x), ∀x∈, (.)
–∂tp(v),q
+aq,p(v)+vp(v),q=y(v) –yd,q
, ∀q∈W,t∈J, (.)
p(v)(T,x) = , ∀x∈. (.)
Letyh(v),ph(v) meet the following system:
∂tyh(v),wh
+ayh(v),wh
+vyh(v),wh
= (f,wh), ∀wh∈Wh,t∈J, (.)
yh(v)(,x) =yh(x), ∀x∈, (.)
–∂tph(v),qh
+aqh,ph(v)
+vph(v),qh
=yh(v) –yd,qh
, ∀qh∈Wh,t∈J, (.)
ph(v)(T,x) = , ∀x∈. (.)
If (y,p,u) and (yh,ph,uh) are the solutions of (.)-(.) and (.)-(.), respectively, then
(y,p) = (y(u),p(u)) and (yh,ph) = (yh(uh),ph(uh)).
We define an elliptic projection operatorRh:W→Whthat satisfies
a(Rhφ–φ,wh) = , ∀φ∈W,wh∈Wh, (.)
and theL-orthogonal projection operatorQ
h:U→Uhthat satisfies
They have the following properties (see e.g. []):
Rhφ–φs≤Ch–sφ, ∀φ∈H(),s= , , (.)
Qhψ–ψ–s≤Ch+s|ψ|, ∀ψ∈H(),s= , . (.)
The following lemmas are very important for a priori error estimates and superconver-gence analysis.
Lemma . For any v∈K,if there exists a constant c> such that
cw≤a(w,w) + (vw,w), ∀w∈W, (.)
then (.)-(.) and (.)-(.) have unique solutions, respectively. Assuming that y(v),p(v)∈H(H),we have
y(v) –yh(v)L∞(L)+y(v) –yh(v)L(H)≤Ch, (.)
p(v) –ph(v)L∞(L)+p(v) –ph(v)L(H)≤Ch. (.)
Proof It follows from (.)-(.) and (.)-(.) that
∂t
y(v) –yh(v)
,wh
+ay(v) –yh(v),wh
+vy(v) –yh(v)
,wh
= ,
∀wh∈Wh,t∈J, (.)
y(v)(,x) –yh(v)(,x) =y(x) –yh(x), ∀x∈, (.)
–∂t
p(v) –ph(v)
,qh
+aqh,p(v) –ph(v)
+vp(v) –ph(v)
,qh
=y(v) –yh(v),qh
, ∀qh∈Wh,t∈J, (.)
p(v)(T,x) –ph(v)(T,x) = , ∀x∈, (.)
Lettingwh=Rhy(v) –yh(v) in (.), we obtain, for anyt∈J,
=∂t
y(v) –yh(v)
,Rhy(v) –yh(v)
+ay(v) –yh(v),Rhy(v) –yh(v)
+vy(v) –yh(v)
,Rhy(v) –yh(v)
. (.)
Applying (.) to (.), we have
d
dty(v) –yh(v)
+cy(v) –yh(v)
≤∂t
y(v) –yh(v)
,y(v) –yh(v)
+ay(v) –yh(v),y(v) –yh(v)
+vy(v) –yh(v)
,y(v) –yh(v)
=∂t
y(v) –yh(v)
,y(v) –Rhy(v)
+ay(v) –yh(v),y(v) –Rhy(v)
+vy(v) –yh(v)
,y(v) –Rhy(v)
. (.)
From (.) and (.), Hölder’s inequality, Young’s inequality with, and Gronwall’s
Lemma . For anyυ,ω∈K,let(y(υ),p(υ))and(y(ω),p(ω))be the solutions of(.)-(.), (yh(υ),ph(υ)),and let(yh(ω),ph(ω))be the solutions of(.)-(.).Then
y(υ) –y(ω)L∞(L)+y(υ) –y(ω)L(H)≤Cυ–ωL(H–), (.)
p(υ) –p(ω)L∞(L)+p(υ) –p(ω)L(H)≤Cυ–ωL(H–), (.)
yh(υ) –yh(ω)L∞(L)+yh(υ) –yh(ω)L(H)≤Cυ–ωL(H–), (.)
ph(υ) –ph(ω)L∞(L)+ph(υ) –ph(ω)L(H)≤Cυ–ωL(H–). (.)
Proof For anyυ,ω∈K, it is clear that
∂t
y(υ) –y(ω),w+ay(υ) –y(ω),w+υy(υ) –y(ω),w
=ω–υ,wy(ω), ∀w∈W,t∈J, (.)
y(υ)(,x) –y(ω)(,x) = , ∀x∈, (.)
–∂t
p(υ) –p(ω),q+ap(υ) –p(ω),q+υp(υ) –p(ω),q
=y(υ) –y(ω),q+ω–υ,qp(ω), ∀q∈W,t∈J, (.)
p(υ)(T,x) –p(ω)(T,x) = , ∀x∈. (.)
Inequalities (.) and (.) follow from the regularity estimates of (.)-(.) and
(.)-(.), respectively. Analogously, we can derive (.) and (.).
Lemma . Let(y,p,u)be the solution of(.)-(.).Assume that u∈L(H).We have
yh(Qhu) –yh(u)L∞(L)+yh(Qhu) –yh(u)L(H)≤Ch, (.)
ph(Qhu) –ph(u)L∞(L)+ph(Qhu) –ph(u)L(H)≤Ch. (.)
Proof It follows from (.) that
Qhu–uL(H–)≤ChuL(H). (.)
Settingυ=Qhuandω=uin (.)-(.), we obtain (.)-(.).
Lemma . Let(y(v),p(v))be the solution of(.)-(.)with v∈K.Suppose that y(v),p(v)∈
H(H).Then the following estimates hold:
Rhy(v) –yh(v)L∞(L)+Rhy(v) –yh(v)L(H)≤Ch, (.)
Rhp(v) –ph(v)L∞(L)+Rhp(v) –ph(v)L(H)≤Ch. (.)
Proof It follows from the definition ofRhand (.)-(.) that
∂t
y(v) –yh(v)
,wh
+aRhy(v) –yh(v),wh
+vy(v) –yh(v)
,wh
= ,
y(v)(,x) –yh(v)(,x) =y(x) –yh(x), ∀x∈, (.)
–∂t
p(v) –ph(v)
,qh
+aqh,Rhp(v) –ph(v)
+vp(v) –ph(v)
,qh
=y(v) –yh(v),qh
, ∀qh∈Wh,t∈J, (.)
p(v)(T,x) –p(v)h(T,x) = , ∀x∈. (.)
Letwh=Rhy(v) –yh(v) in (.). From Hölder’s inequality, Young’s inequality with, and
(.) we derive
d
dtRhy(v) –yh(v)
+cRhy(v) –yh(v)
≤∂t
Rhy(v) –yh(v)
,Rhy(v) –yh(v)
+aRhy(v) –yh(v),Rhy(v) –yh(v)
+vRhy(v) –yh(v)
,Rhy(v) –yh(v)
=∂t
Rhy(v) –y(v)
,Rhy(v) –yh(v)
+vRhy(v) –y(v)
,Rhy(v) –yh(v)
≤C()h∂ty(v)
+C()h
y(v)
+ Rhy(v) –yh(v)
. (.)
Note that
Rhy(v)(,x) –yh(v)(,x) = . (.)
Estimate (.) follows from (.) and Gronwall’s inequality. Similarly, we can obtain
(.).
4 A priori error estimates
In this section, we derive a priori error estimates of the approximation scheme (.)-(.). For ease of exposition, we set
J(u) =
T
y–yd+u
dt,
Jh(uh) =
T
yh–yd+uh
dt.
It is easy to show that
J(u),v=
T
(u–yp,v)dt,
Jh(uh),v
=
T
(uh–yhph,v)dt.
As in [], we assume that there exist neighborhoods of the exact solution uor of the
approximation solution uh inK and a constantc> such that, for anyvorvh in this
neighborhood, the objective functional satisfies the following convexity conditions:
cv–uL(L)≤
J(v) –J(u),v–u, (.)
cvh–uhL(L)≤
Jh(vh) –Jh(uh),vh–uh
Theorem . Let(y,p,u)and(yh,ph,uh)be the solutions of(.)-(.)and(.)-(.).
Suppose that y,p∈H(L)∩L(H).Then
u–uhL(L)≤Ch, (.)
y–yhL∞(L)+y–yhL(H)≤Ch, (.)
p–phL∞(L)+p–phL(H)≤Ch. (.)
Proof It follows from (.), (.), (.), and (.) that
cu–uhL(L)
≤J(u) –J(uh),u–uh
=
T
(u–yp,u–uh)dt–
T
uh–y(uh)p(uh),u–uh
dt ≤ T
(yp,u–uh) + (yhph,uh–u)
+ (uh–yhph,Qhu–u) –
yp–y(uh)p(uh),u–uh
dt = T
(uh–yhph,Qhu–u) +
yhph–y(uh)p(uh),uh–u
dt = T
yp–y(uh)p(uh),Qhu–u
dt+
T
y(uh)p(uh) –yhph,Qhu–u
dt
+
T
(yp,u–Qhu)dt+
T
yhph–y(uh)p(uh),uh–u
dt
:=I+I+I+I. (.)
For the first termI, by the embedding inequalityvL()≤CvH() and Young’s
in-equality withwe get
I=
T
yp–y(uh)p(uh),Qhu–u
dt
≤C()Qhu–uL(L)+Cy(u) –y(uh)
L(H)+Cp(u) –p(uh)
L(H). (.)
By Hölder’s inequality and the embedding inequalityvL()≤CvH()we have
I=
T
y(uh)p(uh) –yhph,Qhu–u
dt
≤CQhu–uL(L)+yh–y(uh)
L(H)+ph–p(uh)
L(H)
, (.)
and the third termIcan be estimates as follows:
I=
T
(yp,u–Qhu)dt
=
T
(yp–QhyQhp,u–Qhu)dt
≤Cy–QhyL(H)+p–QhpL(H)+Qhu–uL(L)
Applying Hölder’s inequality, the embedding inequalityvL()≤CvH(), and Young’s
inequality withtoI, we have
I=
T
yhph–y(uh)p(uh),uh–u
dt
≤C()yh–y(uh)L(H)+ph–p(uh)L(H)
+u–uhL(L). (.)
According to (.), Lemmas .-., and (.)-(.), we obtain
u–uhL(L)≤Ch. (.)
From (.)-(.) and (.)-(.) we have
∂t(y–yh),wh
+a(y–yh,wh) + (uy–uhyh,wh) = , ∀wh∈Wh,t∈J, (.)
y(,x) –yh(,x) =y(x) –Rhy(x), ∀x∈. (.)
Lettingwh=Rhy–yhin (.), we get, for anyt∈J,
∂t(y–yh),y–yh
+a(y–yh,y–yh) +
u(y–yh),y–yh
=∂t(y–yh),y–Rhy
+a(y–yh,y–Rhy) +
yh(uh–u),y–yh
+u(y–yh),y–Rhy
+yh(u–uh),y–Rhy
. (.)
From (.), (.), (.), Young’s inequality with, and Gronwall’s inequality we derive
(.). It is paralleled to get (.).
5 Superconvergence analysis
In this section, we derive the superconvergence between projections of the exact solu-tions and approximation solusolu-tions. Let ube the solutions of (.)-(.). For a fixedt∗
(≤t∗≤T), we divideinto the following subsets:
+=τ:τ⊂,a<ut∗,·<b
,
=τ:τ⊂,ut∗,· τ =aorut∗,· τ =b
,
–=\+∪.
It is easy to see that these three subsets do not intersect with each other and=¯+∪
¯
∪ ¯–. We assume thatuandThare regular such thatmeas(–)≤Ch(see, e.g., []).
Theorem . Let(y,p,u)and(yh,ph,uh)be the solutions of(.)-(.)and(.)-(.),
respectively.Assume that all the conditions in Lemmas.-.are valid and y,p∈L(L∞).
Moreover,we suppose that the exact control,state,and costate solutions satisfy
Then,we have
Qhu–uhL(L)≤Ch
. (.)
Proof Lettingvh=Qhuin (.), we obtain the inequality
T
(uh–yhph,Qhu–uh)dt≥. (.)
It follows from the definition ofQh, (.), and (.) that
cQhu–uhL(L)
≤Jh(Qhu) –Jh(uh),Qhu–uh
=
T
Qhu–yh(Qhu)ph(Qhu),Qhu–uh
dt–
T
(uh–yhph,Qhu–uh)dt
≤
T
u–yh(Qhu)ph(Qhu),Qhu–uh
dt
=
T
(u–yp,Qhu–uh)dt+
T
yh(u)ph(u) –yh(Qhu)ph(Qhu),Qhu–uh
dt
+
T
RhyRhp–yh(u)ph(u),Qhu–uh
dt+
T
(yp–RhyRhp,Qhu–uh)dt
=:I+I+I+I. (.)
For the first term, at timet∗(≤t∗≤T), we have
(u–yp,Qhu–uh) =
+
+
+
–
(u–yp)(Qhu–uh)dx (.)
and
(Qhu–u)|= .
From (.) we get
(u–yp)|+= .
Hence,
I=
T
–(u–yp)(Qhu–uh)dx dt
=
T
–
(u–yp) –Qh(u–yp)
(Qhu–uh)dx dt
≤Ch T
u–yp,–u,–dt
≤Ch T
u–ypW,∞uW,∞·meas–dt
≤Chu–ypL(W,∞)+uL(W,∞)
By using Hölder’s inequalty, the embedding inequalityvL()≤CvH(), and Young’s
inequality,IandIcan be estimated as follows:
I≤C()yh(u) –yh(Qhu)L(H)+ph(u) –ph(Qhu)L(H)
+Qhu–uhL(L) (.)
and
I≤C()Rhy–yh(u)L(H)+Rhp–ph(u)L(H)
+Qhu–uhL(L). (.)
In addition, noting thaty,p∈L(L∞), we have
I≤C()
Rhy–yL(L)+Rhp–pL(L)
+Qhu–uhL(L). (.)
It follows from (.)-(.), (.), and Lemmas .-. that (.) holds for small enough.
Theorem . Let(y,p,u)and(yh,ph,uh)be the solutions of(.)-(.)and(.)-(.),
respectively.Assume that all the conditions in Theorem.are valid.Then
yh–RhyL∞(L)+yh–RhyL(H)≤Ch
, (.)
ph–RhpL∞(L)+ph–RhpL(H)≤Ch
. (.)
Proof From the definition ofRh, (.)-(.), and (.)-(.), for anywhorqh∈Whand
t∈J, we have
∂t(yh–Rhy),wh
+a(yh–Rhy,wh) +
u(yh–Rhy),wh
=∂t(y–Rhy),wh
+u(y–Rhy),wh
+yh(u–Qhu),wh
+yh(Qhu–uh),wh
, (.)
yh(,x) –Rhy(,x) = , ∀x∈, (.)
–∂t(ph–Rhp),qh
+a(qh,ph–Rhp) +
u(ph–Rhp),qh
= –∂t(p–Rhp),qh
+u(p–Rhp),qh
+ph(u–uh),qh
+ (yh–Rhy,qh), (.)
ph(T,x) –Rhp(T,x) = , ∀x∈. (.)
Hence, lettingwh=yh–Rhyin (.), (.) follows from (.)-(.), Hölder’s inequality,
Young’s inequality, Gronwall’s inequality, (.), and (.). Inequality (.) can be similarly
derived.
6 Numerical experiment
In this section, we present a numerical example to validate our superconvergence results. Lett> ,N=T/t∈Z+,t
n=nt,n= , , . . . ,N. Setϕn=ϕ(x,tn) and
dtϕn=
ϕn–ϕn–
By using the backward Euler scheme to approximate the time derivative, we introduce the following fully discrete approximation scheme: find (ynh,pnh–,uhn)∈Wh×Wh×Khsuch
that
dtynh,wh
+aynh,wh
+unhynh,wh
=fn,wh
, ∀wh∈Wh,n= , , . . . ,N, (.)
yh(x) =yh(x), ∀x∈, (.)
–dtpnh,qh
+aqh,pnh–
+unhpnh–,qh
=ynh–ynd,qh
, ∀qh∈Wh,n=N, . . . , , , (.)
pNh(x) = , ∀x∈, (.)
uhn–ynhphn–,vh–unh
≥, ∀vh∈Kh,n= , , . . . ,N. (.)
Let= (, )×(, ),T = ,a= ,b= ., andA(x) be a unit matrix. The following example is solved numerically by a precondition projection algorithm (see e.g. []), where the codes are developed based on AFEPack, which is freely available.
Example The data are as follows:
p(t,x) =sin(πx)sin(πx)( –t),
y(t,x) =sin(πx)sin(πx)t,
u(t,x) =min.,max,y(t,x)p(x,t),
f(t,x) =yt(t,x) –div
A(x)∇y(t,x)+u(t,x)y(t,x),
yd(t,x) =y(t,x) +pt(t,x) +div
A∗(x)∇p(t,x)–p(t,x)y(t,x).
For brevity, we set
|||ϕ|||=
N–l
n=–l
tϕnL()
and
|||ϕ|||=
N–l
n=–l
tϕnH()
,
wherel= for the controluand the statey, andl= for the costatep. In Table , the errors |||Qhu–uh|||,|||Rhy–yh|||, and|||Rhp–ph|||on a sequence of uniformly refined meshes are
listed. It is consistent with our superconvergence results in Section .
Whenh= .e–,t= , andt= ., we plot the profile ofuhin Figure .
7 Conclusions
Table 1 The errors of Example 1
t h |||Qhu – uh||| |||Rhy – yh|||1 |||Rhp – ph|||1
1/10 1.0e–1 9.94516e–2 2.68512e–2 4.72410e–2 1/30 5.0e–2 3.41740e–2 8.46440e–3 1.47458e–2 1/90 2.5e–2 1.20365e–2 2.78014e–3 4.68123e–3 1/270 1.25e–2 3.58914e–3 8.83804e–4 1.50701e–3
Figure 1The numerical solutionuhatt= 0.5 in Example 1.
optimal control problems (see, e.g., [–]). Although bilinear optimal control problems are frequently met in applications, they are much more difficult to handle in comparison to linear or semilinear cases. There is little work on bilinear optimal control problems. Recently, Yang et al. [] investigated a priori error estimates and superconvergence of fi-nite element methods for bilinear elliptic optimal control problems. Hence, our results on bilinear parabolic optimal control problems are new.
Competing interests
Both authors declare that they have no competing interests.
Authors’ contributions
The authors have participated in the sequence alignment and drafted the manuscript. They have approved the final manuscript.
Acknowledgements
The first author is supported by the National Natural Science Foundation of China (11401201), the Hunan Province Education Department (16B105), and the construct program of the key discipline in Hunan University of Science and Engineering. The second author is supported by the scientific research program in Hunan University of Science and Engineering (2015).
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Received: 9 September 2016 Accepted: 8 March 2017 References
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